389
IEEE TRANSACTIONS ON RELIABILITY, VOL. R-31, NO. 4, OCTOBER 1982
On Comparing Estimators of Pr{ Y < X} in the Exponential Case
MSEIR}
Anne Chao
National Tsing Hua University, Hsinchu
-l(_1
k
=
2134
132
02
n('+
Key Words-Exponential distribution, Maximum likelihood
estimator, Minimum variance s-unbiased estimator.
-2f4
+
Reader AidsPurpose: Report a derivation
Special math needed for explanations: Statistics
Special math needed to use results: Same
Results useful to: Statisticians, Reliability theoreticians
(1 +
Abstract-Let X and Y be s-independent exponentially distributed
random variables with mean P and a respectively. This work povides simple approximations for s-bias and mean square error of the maximum
likelihood estimator of Pr{ Y < X} for two cases: 1) both a and p are
unknown; 2) only P is unknown. When a is known, the mean square error is compared with that of the minimum variance s-unbiased estimator
and a preference relationship between them is established using the mean
square error criterion.
1. INTRODUCTION
The problem of estimating Pr{ Y < X} has been extensively studied in reliability and related fields. It arises when
a component of strength X is subjected to a stress Y. The
component fails when and only when X < Y. If X and y
are s-independently distributed as exponential random
variables with mean P and a respectively, we have reliability
R
=
Pr{Y<X} = P1/(a + P1).
O13
+
iS:
2P4
+
2
-2
-1
435
+
(l+
)
8P5
-
+
k-)
k (3a)
+(3 )2/(
4136
(+
)
n
6136 - 28137 + 8R8
-4
O(n-5).
(3b)
The maximum likelihood estimator (MLE) when a = 1
R = X/(1 +
X).
(4a)
Both estimators were compared [5] using 15-point
Gaussian Laguerre quadrature to obtain approximate
MSE and s-bias of the MLE. For large n, the precision of
this quadrature was not sufficient for accurate calculation
of the first two moments of the MLE. Therefore, it seems
worthwhile to provide simple and satisfactory approximation formulas for s-bias and MSE of the MLE, although
bounds were obtained [7]. Section 2 derives such formulas.
The MLE and MVUE are also compared.
If both a and P are unknown, and Y1, Y2, ..., Yn is a
random sample from Y, then the MLE of R is:
(la) R = X/(X + Y),
Y= 2 Y/fn.
(4b)
The exponential case has been discussed in [1, 4, 5, 7 - 9]. Ref. [7] also developed bounds for s-bias of R in this case.
We first assume that one of the parameters, say a, is However, the suggested bounds did not have explicit exknown and, without losing generality, let a = 1. In this pressions. Section 3 presents corresponding suitable approximations.
case,
R
=
P/(1 + ).
2. APPROXIMATIONS WHEN a = 1
(lb)
Let f(x)
Suppose X1, X2, ...., X, is a random sample chosen from
-x/(l
+
x).
distribution of X. The minimum variance s-unbiased
R - R = f(X) - f(13)
estimator (MVUE) P of R is [8]:
8
(5)
= E fli(13(X - 13)i/i! + r8,
(n - 1)!( -1)n-' n-i ( -1l)(nxi)'
=
pi-nxl
l
-n)- =5
R~~~~~
(2)
fli'(X) -( - 1)'+1i!(1 + x)1~'i+l, i-th derivative of f(x),
,
X Xe/n.
X- i=l
r8 -JR9(u)(X - 13)/9!, is aremainder,
Using a result in [2], [5] showed that the mean square error u is a number between X and (3. The central moments of X
are:
(MSE) of P is:
0018-9529/82/1000-0389$OO.75 © 1982 IEEE
390
IEEE TRANSACTIONS ON RELIABILITY, VOL. R-31, NO. 4, OCTOBER 1982
E{X - [3}
0,
=
fli) = fli)(q3)
E{(X - (3)2} = (22n',
E{(X
-
[3)3}
The asymptotic MSE of the MLE is:
(2[33)n 2,
=
_4[33
[ 32
E{(X
-
4)} = (334)nw2
E{(X
-
)5} = (20O35)n 3
(6I43)n-3
+
+
(6)
MSE{R}
)4-n-l
-
=
+
-96[35 + 362(36 - 248[37 + 29(8
-6P4
+ (5040[38)n7.
8[5
+
-
(I + °
24P5
[36
-
+
58(36
+
P)
+
(1[
22(37
[38
-
(7)
s-efficiencyrespectively.
R
R
.5
3
1.0
.75
3.0
.90
9.0
.237E-1 .288E-1 .183E-1
.95
19.0
.99
99.0
n = 5
10
.122E-1
.143E-1
.862E-2
.490E-2
.108E-2
50
100
.249E-2
.125E-2
.282E-2
.141E-2
.164E-2
.815E-3
.917E-3
.455E-3
.200E-3
.990E-4
dif.
0
0
0
1
0
.107E-1
.242E-2
.495E-2 .566E-2 .332E-2 .186E-2 .408E-3
- [3}
1)f 3)
[f2)12
+
4
fI)f4)
31]2
36
+
+
)
+
*E{(X +
-
max
E{(X-[3)4}
3
3)ft 6)
360
max dif. is the maximum difference with results of [5] in the third
1)ft5)
60
l_57576
+ J 2)f 5)
120
+
E implies an exponent to base 10.
significant digit.
E{(X - [)}
ft),f(7)
~2520
t360tS }, 8{XO(n~5),
I
+
ft2)ft4)
24
f
[)6}
- 7}
[3(X[4)]2
+
+
2)ft3)
6
+
+~
+
now compared with those provided in [5], in which
15-point quadrature was used to approximate the first two
moments of P. Tables l and 2 compare s-bias and
TABLE 1.
UR1)]2 E{(X -_ o2}
t)f2)]
2)]25
E{E{(R\R)}} -- [ft)]2
(R - R)2
E{(X [3)2} ++ [Pfl)f
+
(9)
Negative s-bias of R using (7).
We proceed to find the approximate MSE. From (5), we have
E(X
El
ni
If the O(n-5) terms in (7) and (9) are ignored, we then
obtain approximate s-bias and MSE of R. The results are
n_3
O(n5).
+
°)
+ O(n-5).
E{r8} = O(n5). Take s-expectation on both sides of (5)
and substitute the preceding moments.
XV
P2
2P3 - 04
ERI - R = (1+ )3 n + (1 + f)
+
(1 +
= (105P8)n 4 + (2380P8)n 5
(7308P8)n-6
fl-
6
n+3
(24Pf5)n-4,
E{(X - )7} = (210f37)n 4 + (924[37)n 5 + (720P7)n 6,+
)8}
5[34
+
(1 + [3)
(1 + [3
18[34 - 44(35 + 13(36
+
(1 + [3)
E{(X - 13)6} = (15(36)n 3 + (130(6)n 4 + (120[36)n5,
E{(X -
+
+ ft
+
f
72
f2)Jf6)
720
(8)
The results in tables 1 and 2 are very close to those of
[5]. As anticipated, the proposed formulas are better when
the sample size becomes large. Even for small sample sizes,
the formulas still perform satisfactorily although the
results are derived asymptotically. For large n, computational difficulty arises in calculating the 15-point GaussLaguerre quadrature [5] and for R < 0.5, n > 50, the
~~~~~~~~~~~quadrature is not accurate at all. Apparently, the proposed
formulas do not have those drawbacks. Numerical results
further show that the values of (7) are precisely between
bounds obtained in [7, p 411.
A similar method has also been applied [3] to obtain
the MSEs of estimators of reliability for k-out-of-rn
We now compare both estimators based on MSE
criterion. Using (3b) and (9), we can find for any fixed n
CHAO: ON COMPARING ESTIMATORS OF Pr(Y.LT.X) IN THE EXPONENTIAL CASE
which implies [6, p 153] that nl'2(R - P) i 0. Hence we
TABLE 2
Mean square s-efficiency of R relative to R
(MSE{R}/MSE{R}) using (3b) and (9).
R
.5
(3
1.0
3.0
9.0
.936
1.111
1.379
n =
5
10
25
50
100
.75
19.0
.99
99.0
An advantage of the proposed approximation method
(besides the simplicity) is that we can easily get more accurate results just by retaining more terms in the expansion
procedures.
3. APPROXIMATIONS WHEN a IS UNKNOWN
1.051
1.025
1.059
1.029
The method introduced in the previous section can be
extended in an obvious way: Let g(x, y) x/(x + y),
3
105
R - R = g(X, Y) - g(p, or)
1.201
1.082
1.260
1.102
0
4
12
1.041
1.021
estimators.
1.657
1.520
1.075
1.035
1.018
1.009
have established the asymptotic equivalence of both
.95
.971
.989
.995
.997
max dif.
.90
391
1.314
1.120
Other than n = 5, the max dif is 26.
max dif. is the maximum difference with results of [5] in the third
significant digit.
8
(
_ 7
i=1
ig(X, Y)
j=o a\ Xj3yi
x=p,y=a
the approximate interval of P (or of R) for which the MLE
(X- ) (Y(10)
+ ,(
has smaller MSE. Such intervals are given in table 3 for
i!(i 9 / 9
y) \
several sample sizes. In other words, the mean square
(X -(5)9(Ys-efficiency ofRrelative toP (MSE{P}/MSE{P}) ( 1 if Z( =
\j= axiaY9 / =
9!(9 (or R) is located in the interval as listed. Otherwise, the
MVUE is more s-efficient.
w is between X and (, v is between Y and a. Using the
moments of X, Y and applying similar method as before,
we have asymptotic s-bias:
TABLE 3
Intervals where the MLE has smaller MSE using (3b) and (9)
e(e
+
(1
2
f2
n
interval of 0
interval ofRR
(0, 1.67)
(0, .625)
5
(0, 1.48)
(0, 1.43)
(0, 1.40)
10
15
20
(0, 1.37)
(0, 1.36)
(0, 1.36)
(0, 1.34)
30
40
50
100
=
+
(0, .597)
(0, .588)
(0, .583)
(0, .578)
(0, .576)
(0, .576)
(0,.573)
Q- 14Q3
9)
+
30Q2
+
Q'
-
36Q'
-
Q)4
14Q
+ 1
n-3
+ 207Q4 - 352Q3 + 207Q2 - 36Q + 1
+
+ O(n5)
Q
4
Q)
(11)
a/fl.
As n becomes large, we can ignore the n3, n-4 terms in Ref. [7, p 43] shows that the s-bias and MSE of R satisfy(3b) and (9). Thus it is obvious that the MLE has smaller
M {, =
s
MSE i.f.f.-Q
n(1 + Q)
MSE{R} = Bias{R}[ 1 ]
1 +
_4fl3
+
5(P4
2(4, or
(3 e 4/3, or R 4/7.
e
That is, the intervals of (3 in table 3 will converge to (0, 4/3)
as n increases. The results are generally consistent with [5].
From (3b), (7), (9), we can show that
n'[- R]
= 0(w"l2) O~ ,
~~~~+
~~~~~
R)}
=
Var{n"'2(R
0(n-') -> 0,
+ (+
e
Q
Q
)2
(12)
Consequently,
2Q2
M
MS{R} = (I1+)e)
492(29 - l)(e - 2)
(I1+e)6
l2&2(2Q4 - 1793 + 3292 _ 17p + 2)
(1
+
e)S
n-3
+
O(n~4).
(13)
IEEE TRANSACTIONS ON RELIABILITY, VOL. R-31, NO. 4, OCTOBER 1982
392
REFERENCES
We now examine the performance of (11). Table 4 lists
the negative s-bias for several values of n and Q. It is suffiA.M. Beg, "Estimation of Pr{ Y < X} for exponential family,"
7] which
which[1]
<QQ < 1 [7]
onlyfor
to givevlues
for 0 <
calculated
only
cient to give values
IEEE Trans. Reliability, vol R-29, 1980, pp 158-159.
bounds of s-bias for some sample sizes > 10. Almost all [2] B.J.N. Blight, P.V. Rao, "The convergence of Bhattacharyya
bounds," Biometrika, vol 61, 1974, pp 137-142.
values are between the bounds provided in [7], and (11) is
[3] A. Chao, "Approximate mean squared errors of estimators of
quite satisfactory even for small n.
[4]
4
TABLE
TABLE.4
Negative s-Bias of R using
[51
(I11).''
n
5
10
25
50
100
Q = .01
.236E-2
148E-1
.189E-1
.187E-1
.166E-1
.138E-1
.138E-1
.108E-1
.783E-2
.500E-2
.239E-2
.106E-2
.709E-2
.938E-2
.947E-2
.854E-2
.854E-2
.716E-2
.563E-2
.409E-2
.262E-2
125E-2
.399E-3
.276E-2
.373E-2
.381E-2
.347E-2
.347E-2
.292E-2
.231E-2
.168E-2
.108E-2
.515E-3
.196E-3
137E-2
.186E-2
.191E-2
.174E-2
.174E-2
.147E-2
.1 16E-2
.847E-3
.543E-3
.260E-3
.970E-4
.680E-3
.927E-3
.955E-3
.873E-3
.873E-3
.1
.2
.3
.4
.5
.5
.6
.7
.8
.9
[61
[7]
[8]
[9]
.738E-3
.584E-3
.426E-3
.273E-3
131E-3
Anne Chao; Institute of Applied Mathematics; National Tsing Hua
University; Hsinchu; TAIWAN, Rep. of China.
Anne Chao is an Associate Professor, Institute of Applied
Mathematics, National Tsing Hua University, Taiwan. She received the
PhD degree in Statistics from University of Wisconsin-Madison in 1977.
Her interests are statistical inference and occupancy problems. She is a
member of Institute of Mathematical Statistics and American Statistical
Association.
Manuscript TR81-96 received 1981 September 5; revised 1982 March 1.
I thank the Editor for helpful comments, and a referee
for pointing out a recent paper and many valuable suggestions.
. . . . . . . . .
"Analysis of instability induced by secondary slow trapping in
scaled CMOS devices", Masaharu Noyori O Semiconductor Research Laboratory E Matsushita Electric Industrial Co., Ltd. o
3-15 Yakumonakamachi E Moriguchi, Osaka 560 JAPAN.
(TR82-75)
"An algorithm for obtaining simplified prime implicant sets in
fault-tree and event-tree analysis", Makoto Inomata, Mgr. Pers.
Group El Public Relations Dept. (Copy ff82106) 0 Hitachi, Ltd.
w Nippon Bldg. 6-2 0 Otemachi 2-chome, Chiyoda-ku E Tokyo
100 JAPAN. (TR82-76)
"Forced frequency-balancing technique for discrete capacity systems", Dr. C. Singh O Dept. of Electrical Engineering O Texas
A&M University O' College Station, TX 77843 USA. (TR82-77)
"Fault-tree analysis applied to software", Dr. Nancy G. Leveson
El Information & Computer Science El1 University of California Ol
Irvine, CA 92717 USA. (TR82-78)
"A fast efficient algorithm for solvinig the MLEs of the 3-parameter uncensored gamma distribution", Dr. G. P. Boicourt El1
Accelerator Technology Division Ol Los Alamos National Laboratory Ol Los Alamos, NM 87545 USA. (TR82-83)
X}
AUTHOR
ACKNOWLEDGMENT
M anuscripts Received
reliability for k-out-of-m systems in the independent exponential
case," J. Amer. Statist. Assoc., vol 76, 1981, pp 720-724.
N.L. Johnson, "Letter to the Editor," Technometrics, vol 17, 1975,
p 393.
G.D. Kelley J.A. Kelley, W.R. Schucany, "Efficient estimation of
Pr{ Y < X} in the exponential case," Technometrics, vol 18, 1976,
pp 359-360.
C.R. Rao, Linear Statistical Inference and Its Applications, second
edition, John Wiley, New York, 1973.
Y.S. Sathe, S.P. Shah, "On estimating P{X > Y} for the exponential distribution," Comm. Statist. A, vol A10(1), 1981, pp 39-47.
in the exponenH. Tong, "A note on the estimation of Pr{ Y <
tial case," Technometrics, vol 16, 1974, p 625, Errata, vol 17, 1975,
p 395
p 395
H.
Tong, "On the estimation of Pr{ Y K X} for exponential
families," IEEE Trans. Reliability, vol R-26, 1977, pp 54-56.
For information, write to the author at the listed address.
"Reliability of optical systems for data transmission", Prof. Dr.
D. f. Lazaroiu El Bd. Hristo Botev 8, E0 R - 70 335 Bucharest El
ROMANIA. (TR82-79)
"An approach to quantifying common-cause failure in system
analysis", Gareth W. Parry L NUS Corp. E 910 Clopper Road
O Gaithersburg, MD 20878 USA. (TR82-80)
"Optimal sequencing of items in a consceutive-2-out-of-n system", Dr. F. K. Hwang F] Bell Laboratories (2C379) El 600
Mountain Avenue O Murray Hill, NJ 07974 USA. (TR82-81)
"Erlang distribution - A graphical method", Slavko Sever n Siget
18A/VIII o 41000 Sabreb, YUGOSLAVIA. (TR82-82)
~~~~~~~~~~~~~"Automatically controlled
2-vessel pressure cooker test", Kiyoshi
Ogawa El Device Rel. Sec., Int. Electronics Dev. Div. Ol Musahsino Electrical Communication Laboratory El Nippon Telegraph &
Telephone Public Corp. El 3-9-11, Midoricho, Musashino>-shi,
180 JAPAN. (TR82-84)
~~~~~~~~Tokyo
"Analysis of selected failures in en-route air traffic control system", Bharat Bhargava El Dept. of Computer Science El Univtersity of Pittsburgh El Pittsburgh, PA 15260 USA. (TR82-85)